Provide proof that the modified choice issue has an NP-complete solution; Is there a spanning tree with a graph G and a target cost c in which the maximum feasible payment at any vertex does not exceed the target cost?

Given a directed graph with positive edge lengths (weights) and two distinct vertices u and v in the graph, the “all-pairs u-constrained v-avoiding shortest path problem” is the problem of computing for each pair of vertices i and j the length of the shortest path from i to j that goes through the vertex u and avoids vertex v. If no such path exists, the answer is ∞. Describe an algorithm that takes a graph G = (V, E) and vertices u and v as input parameters and computes values L(i,j) that represent the length of u-constrained v-avoiding shortest path from i to j for all 1 ≤ i, j ≤ |V |, i ̸= u, j ̸= u, i ̸= v, j ̸= v. Prove your algorithm correct. Your algorithm must have running time in O(|V |2). Detailed pseudocode is required.

Provide evidence that the modified version of the choice problem has an NP-complete solution; Exists, given a graph G and a target cost c, a spanning tree in which the highest possible payment at any vertex does not exceed the target cost?

Problem 4: Consider the connected simple graph G given below. k Graph G 1. Use depth-first search to produce a spanning tree for the simple graph G. Choose vertex a as the root of this spanning tree. You must show how the spanning tree is constructed step by step as you add vertices and edges. Otherwise, your answer is wrong. Show your work step by step. 2. Use breadth-first search to produce a spanning tree for the simple graph G. Choose vertex a as the root of this spanning tree and assume that all vertices are ordered alphabetically. You must show how the spanning tree is constructed step by step as you add vertices and edges. Otherwise, your answer is wrong. Show your work step by step. 3. Is a spanning tree of a simple connected graph unique? Explain your answer clearly.